The idea of higher categories and the corresponding higher topos theory is an old one, going back to the ideas of Grothendieck who started to pursue it. For many years, though, it kept being pursued without finding a generally useful form.

This is changing now. For several years André Joyal amplified the fact that the weak Kan complexes introduced by Boardman and Vogt, which he started calling quasi-categories, are a model for (infinity,1)-categories for which a good comprehensive closed theory can be obtained, that completely parallels and generalizes ordinary category theory. Based on these ideas by Joyal and work by people like Carlos Simpson, more recently Jacob Lurie presented a comprehensive textbook on the subject

Idea of this Journal Club

The general patterns will be that concepts in quasi-categories are relatively easily understood as relatively straightforward generalizations of the corresponding familiar 1-categorical concepts, but that concrete computations in concrete models may tend to be a bit more demanding.

There are mainly four different concrete realizations of the notion (∞,1)(\infty,1)-category, which can all be related to each other and each have advantages and disadvantages for certain purposes:

the construction principle of complete Segal spaces lends itself best to iteration, which then yields models for (infinity,n)-categories for higher nn: ∞\infty-categories in which only the kk-morphisms for k>nk \gt n are required to be invertible.

Segal categories are a weak form of simplicial categories. They have been frequently used by Carlos Simpson and Betrand Toën in their work.

Vogt’s main theorem involved a category of homotopy coherent diagrams defined on a topologically enriched category and showed it was equivalent to a quotient category of the category of (commutative) diagrams on the same category.